If I have persistent ankle pain when running should I take a break?

Yes! See a doc! Ankle pain during running could be from many conditions including tendinitis, a stress fracture, ankle instability, arthritis, etc. See a sports medicine specialist who has an interest in running injuries. Consider a trial of painless swimming, biking or use of the elliptical to minimize the impact on your ankle. Update your athletic shoes if over a year old or if they have over 300 miles on them.
Yes. The key is to sort out where the pain is coming from. And then tailor treatment to exact problem. Tendonitis will need rest and strengthening, a stress fracture needs rest, instability needs strengthening. Other problems may require more complex treatment after an evaluation by an orthopedist.